The daily sales of a retail store are normally distributed with a standard deviation of $500. If the probability that sales will be less than $10,000 in any one day is 40%, what are the mean daily sales?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.40) and its Z score.
-Z = (score-mean)/SD = (10,000-mean)/500
$10,126.65
To find the mean daily sales, we need to use the properties of the normal distribution.
Let X be the daily sales of the retail store.
We know that X follows a normal distribution with a standard deviation of $500.
The probability of sales being less than $10,000 in any one day is 40% or 0.40.
To find the mean daily sales, we need to find the z-score corresponding to the probability of 0.40. We can then use the z-score formula to calculate the mean.
The z-score formula is given by:
z = (X - μ) / σ
Where:
X = $10,000 (value)
μ = mean daily sales (unknown)
σ = standard deviation = $500
Step 1: Find the z-score.
To find the z-score, we use the standard normal distribution table or a calculator. The z-score corresponds to the cumulative probability of 0.40 is approximately -0.253.
Step 2: Rearrange the z-score formula to solve for μ.
z = (X - μ) / σ
Rearranging, we get:
μ = X - (z * σ)
Substituting the given values, we have:
μ = $10,000 - (-0.253) * $500
μ = $10,000 + $126.5
Therefore, the mean daily sales is $10,126.5.
To find the mean daily sales, we need to use the concept of z-scores and the standard normal distribution.
Let's denote the mean daily sales as μ and the standard deviation as σ. We know that the standard deviation is $500.
First, we need to find the z-score corresponding to a probability of 40%. The z-score can be calculated using the standard normal distribution table or a calculator. For a probability of 40%, the z-score is approximately -0.253.
The formula for converting a value to a z-score is:
z = (x - μ) / σ,
Where:
z = z-score,
x = value,
μ = mean,
σ = standard deviation.
In this case, we know x = $10,000, z = -0.253, and σ = $500. Plugging these values into the formula, we can solve for μ:
-0.253 = (10,000 - μ) / 500.
Multiply both sides of the equation by 500:
-0.253 * 500 = 10,000 - μ.
-0.253 * 500 = -126.5, so the equation becomes:
-126.5 = 10,000 - μ.
Rearrange the equation to solve for μ:
μ = 10,000 + 126.5.
After evaluating the equation, we find that the mean daily sales will be approximately $10,126.5.
Therefore, the mean daily sales of the retail store is approximately $10,126.5.